The Aumann-Serrano performance index for multi-period gambles in stock data

Hodoshima, Jiro; Yamawake, Toshiyuki Article The Aumann-Serrano performance index for multi-period gambles in stock data Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Hodoshima, Jiro; Yamawake, Toshiyuki (2020) : The Aumann-Serrano performance index for multi-period gambles in stock data, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 13, Iss. 11, pp. 1-18, https://doi.org/10.3390/jrfm13110288 This Version is available at: https://hdl.handle.net/10419/239339 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Journal of Risk and Financial Management Article The Aumann–Serrano Performance Index for Multi-Period Gambles in Stock Data Jiro Hodoshima * and Toshiyuki Yamawake Faculty of Economics, Nagoya University of Commerce and Business, 4-4 Sagamine, Komenoki-cho, Nisshin-shi, Aichi 470-0193, Japan; [email protected] *Correspondence: [email protected]; Tel.: +81-561-73-2111; Fax: +81-561-73-1202 Received: 27 October 2020; Accepted: 17 November 2020; Published: 20 November 2020   Abstract: We present an empirical study of the Aumann-Serrano performance index for multi-period gambles when the underlying stochastic process is assumed to be a normal mixture process with time-varying volatility. We compare the Aumann-Serrano performance index for multi-period gambles with that for one-period gambles as well as the Sharpe ratio. Our empirical study is obtained using a selection of U.S. stock data and shows evaluation of a selection of stocks becomes more distinct in multi-period gambles than in one-period gambles in the sense that a favorable evaluation score becomes even better in multi-period gambles than in one-period gambles while an unfavorable evaluation score becomes even worse in multi-period gambles than in one-period gambles. Keywords: Aumann-Serrano performance index; multi-period gamble; Sharpe ratio; stock data JEL Classification: G11; C22; C46 1. Introduction Providing appropriate performance measures is quite important in finance since the evaluation of assets, projects, cash flows, etc. is essential in finance via appropriate performance measures. Recently, Aumann and Serrano (2008) proposed the economic index of riskiness (hereafter the AS index) based on an axiomatic approach. While the AS index is an index of risk, it has been used in performance measures. A performance measure based on the AS index was proposed by Kadan and Liu (2014) , which is obtained directly from the AS index, i.e., the reciprocal of the AS index. We use the performance measure based on the AS index proposed by Kadan and Liu (2014) to evaluate financial assets. In this study, we intend to study multi-period gambles which stand for gambles over time where gambles stand for random variables with uncertain outcomes representing asset returns, projects, cash flows, etc. There exist many occasions where it is more appropriate to treat gambles as multi-period gambles rather than one-period gambles. For example, when we invest in financial assets, we are often concerned with long-run profits rather than short-run profits. If asset returns are observed daily and we are interested in profits five years later, then it may be more appropriate to treat daily observations as realizations of multi-period gambles rather than those of one-period gambles since we can take into account dynamic properties of returns more naturally in multi-period gambles compared to one-period gambles where returns in the future are properly discounted in multi-period gambles but not in one-period gambles . To the best of our knowledge, empirical studies of multi-period gambles seem to be rare1although providing a theoretical framework of multi-period gambles may not be rare. We use a setup considered by Kadan and Liu (2014) to deal with multi-period gambles based on the axiomatic approach of Aumann and Serrano (2008). In other words, we consider a T-period J. Risk Financial Manag. 2020,13, 288; doi:10.3390/jrfm13110288 www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2020,13, 288 2 of 18 gamble g= (g1 , g2 , ··· , gT) whose AS performance index PAS(g) is given by the unique solution of the implicit equation T ∑ t=1 ρt−1E[exp(−PAS(g)·gt)] = T ∑ t=1 ρt−1, where ρ∈( 0, 1 ) denotes a discount factor. The above equation is equal to Equation (8) of Kadan and Liu (2014) . To the best of our knowledge no studies of the above AS performance index PAS(g) for multi-period gambles were conducted since it was proposed by Kadan and Liu (2014). We shall make an empirical study of the above AS performance index using a selection of U.S. stock market data. We consider as multi-period gambles a stochastic process given by a normal mixture process with time-varying volatility. Please note that one must assume some models of the dynamic process of g since the implicit equation given above involves expectation of gt at each time t in order to obtain the AS performance index PAS(g) for multi-period gambles. We remark that a normal mixture process is a process with its distribution each time being independently and identically distributed (i.i.d.). A normal mixture process is a simple stochastic process but is known to capture well characteristics of distributions often observed in financial data such as skewed and heavy tail distributions as well as symmetric and multi-modal distributions. We incorporate time-varying volatility into a normal mixture process in this study. Time-varying volatility is another stylized feature often observed in financial data. We use a family of generalized autoregressive conditional heteroskedastic (GARCH) models proposed by Bollerslev (1986), which is a generalization of the autoregressive conditional heteroskedastic (ARCH) model originally proposed by Engle (1982), as a time-varying volatility model. By combining a normal mixture process with time-varying volatility, we can create state-dependent volatility models which allow different volatility responses to stock price shocks in stable and stressful market conditions, which is not possible under single-state GARCH models allowing only one mechanism of volatility to market shocks. State-dependent volatility models are more flexible to capture characteristics of financial data than single-state volatility models. When we use a family of GARCH models for a selection of U.S. stock market data, we try to take into account the so-called leverage effect of negative returns with more influence on the underlying volatility than positive returns. When we evaluate multi-period gambles, we should capture well the underlying process of multi-period gambles to be used in the performance measure. Otherwise we cannot obtain the relevant evaluation. Therefore, modeling appropriately the underlying stochastic process is crucial to conduct evaluation based on the performance measure for multi-period gambles. Since providing proper evaluation for financial assets, projects, cash flows, etc. is essential to investors, financial managers, and regulators, our approach should be very useful to them. We compare performance of a selection of U.S. stocks when they are evaluated as realizations of multi-period gambles with that when they are evaluated as realizations of one-period gambles. Performance of one-period gambles is computed by the AS performance index for one-period gambles, i.e., the solution PAS(g)of the implicit equation E[exp(−PAS(g)g)] = 1 for one-period gambles g , assuming the underlying stochastic process of gambles g to be a normal mixture process with time-invariant volatility, a special case of the normal mixture process with time-varying volatility of GARCH families, i.e., the assumption of the stochastic process of multi-period gambles in this study. The difference of the assumption of the underlying stochastic process of multi-period gambles from that of one-period gambles is whether the volatility process is time-varying or time-invariant. Therefore, we explicitly model time-varying volatility as GARCH families in evaluation of multi-period gambles with late return observations properly discounted. On the other hand, we assume volatility to be time-invariant in evaluation of one-period gambles with late return observations not discounted. We will examine the consequences of the difference in the model of J. Risk Financial Manag. 2020,13, 288 3 of 18 multi-period gambles from that of one-period gambles in this study. The data we use in this study covered the global financial crisis where tremendous volatility clustering was observed in stock markets. Therefore, we anticipate in our sample period the AS performance index for multi-period gambles is more relevant than that for one-period gambles since the former takes into account volatility clustering but the latter does not. In addition, our anticipation is verified by our empirical results. Our results show when stocks are more poorly evaluated by the AS performance index for one-period gambles than the Sharpe ratio, then they are evaluated even more poorly in multi-period gambles than in one-period gambles. On the other hand, when stocks are more favorably evaluated by the AS performance index for one-period gambles than the Sharpe ratio, then they are evaluated more favorably in multi-period gambles than in one-period gambles. In other words, evaluation of a selection of stocks becomes more distinct in multi-period gambles than in one-period gambles in the sense that a favorable evaluation score becomes even better in multi-period gambles than in one-period gambles while an unfavorable evaluation score becomes even worse in multi-period gambles than in one-period gambles. The rest of the paper is organized as follows. In Section 2, we present the literature review related to our method. In Section 3, we present our setup and the assumptions of the underlying stochastic process. In Section 4, we show our empirical study using a selection of U.S. stock data. In Section 5, we present concluding comments. 2. Literature Review In this section, we provide the literature review related to our study. The AS index has received a lot of attention in finance (see, e.g., Foster and Hart 2009;Hart 2011; Homm and Pigorsch 2012a ; Kadan and Liu 2014 ;Sculze 2014; Niu et al. 2018 ; Hodoshima and Miyahara 2020 ). Another performance measure based on the AS index in the literature is the economic performance measure proposed by Homm and Pigorsch (2012a) for one-period gambles, which is a modification of the Sharpe ratio by replacing standard deviation by the AS index. Hodoshima (2020) compared the economic performance measure proposed by Homm and Pigorsch (2012a) and the AS performance index for one-period gambles proposed by Kadan and Liu (2014) and showed the latter is superior to the former. There also exist plenty of other performance measures. See, e.g., Cogneau and Hubner (2009a,2009b) , and Cherny and Madan (2009). While these performance measures were compared by many studies, these preceding studies were mainly analyzed as static performance measures, i.e., the underlying gambles are assumed to be one-period gambles. Also many of these performance measures are proposed in ad hoc ways. On the other hand, the performance measure based on the AS index is based on an axiomatic approach. Although performance measures based on the AS index have been used to evaluate performance of various funds, portfolios, individual assets, etc. (see, e.g., Homm and Pigorsch 2012a; Kadan and Liu 2014;Hodoshima 2019;Hodoshima and Otsuki 2019), they are confined to only one-period gambles. Therefore, our use of the AS performance index for multi-period gambles in empirical studies is the first trial in the literature. A normal mixture process or a normal mixture distribution has been studied in the statistics and econometrics literature for several decades. See, e.g., Everitt and Hand (1981); McLachlan and Peel (2000) ; and Titterington et al. (1985) . There are also many applications of the normal mixture process in finance (cf., e.g., Kon 1984;Alexander 2004;Hodoshima 2019). To capture the leverage effect, we employ the asymmetric GARCH model (hereafter AGARCH model) considered by Engle (1990) and Engle and Ng (1993) and the model proposed by Glosten et al. (1993) (hereafter GJR model). Therefore, we use a normal mixture process with time-varying GARCH volatility taking into account the leverage effect as the underlying stochastic process for multi-period gambles. Models we consider in this paper are similar to those treated in Alexander and Lazar (2009). However, we use an estimation method different from Alexander and Lazar (2009) who used the maximum likelihood estimator (MLE). In other words, we use an empirical characteristic function approach due to Xu and Wirjanto (2010) as an estimation method for our models. In particular, we estimate a normal J. Risk Financial Manag. 2020,13, 288 4 of 18 mixture process with components up to three, which is different from the previous empirical study of Alexander and Lazar (2009) , where components in the normal mixture process were restricted to two. It was stated in Alexander and Lazar (2009) that in estimating a normal mixture process with three components by the MLE, convergence was harder to attain. On the other hand, we do not encounter such problems in our empirical study by an empirical characteristic function approach. 3. The Model for Multi-Period Gambles In this section, we present the theoretical framework for multi-period gambles and the normal mixture process with time-varying GARCH volatility as the process of multi-period gambles. As shown by Kadan and Liu (2014), our theoretical framework for multi-period gambles is given as follows. We consider a T-period gamble g= (g1 , g2 , ··· , gT) where each gt denotes a gamble at time t(t= 1, 2, ··· , T) . We consider an investor with a time-separable utility function U:RT7−→ R with the form given by U(w1,w2,··· ,wT) = T ∑ t=1 ρt−1u(wt)(1) where u denotes a utility function, ρ∈( 0, 1 ) is a discount factor, and (w1 , w2 , ··· , wT) denotes the level of wealth from 1 to T. Notice domain of the level of wealth is not, unlike Kadan and Liu (2014), restricted to positive region since the AS performance index can be defined in positive region as well as negative region (cf. Hodoshima and Miyahara 2020). First we present two definitions to provide ordering of one-period gambles. Definition 1 (Definition 2 of Kadan and Liu (2014)) . A gamble g is wealth-uniformly rejected by an investor with utility function u,if u rejects gat all initial wealth levels w0. Definition 2 (Definition 3 of Kadan and Liu (2014)) . A gamble g wealth-uniformly dominates a gamble g0 if whenever gis wealth-uniformly rejected by a utility function u,g0is also wealth-uniformly rejected by u. Then, the following proposition provides a property of the AS performance index for one-period gambles. Proposition 1 (Proposition 1 of Kadan and Liu (2014)) . Wealth-uniform dominance induces a complete order on the set G of one-period gambles. This order can be represented by a performance index PAS(g) assigned to any gamble g∈ G,which is given by the unique solution to the implicit equation E[exp(−PAS(g)g)] = 1. (2) That is, for any two gambles gand g0,gwealth-uniformly dominates g0if and only if PAS(g)≥PAS(g0). The property of the above proposition is extended to multi-period gambles as follows. Proposition 2 (Proposition 3 of Kadan and Liu (2014)) . Wealth-uniform dominance induces a complete order on GT .This order can be represented by a performance index PAS(g) assigned to any T-period gamble g= (g1,g2,··· ,gT)∈ GT,which is given by the unique solution to the implicit equation T ∑ t=1 ρt−1E[exp(−PAS(g)·gt)] = T ∑ t=1 ρt−1. (3) That is, for any two gambles gand g0,gwealth-uniformly dominates g0if and only if PAS(g)≥PAS(g0). There have been no empirical studies of multi-period gambles yet so that we do not know what kind of consequences the concept of multi-period gambles entails. In this study, we use a selection of J. Risk Financial Manag. 2020,13, 288 5 of 18 U.S. stocks as examples of multi-period gambles to compute the AS performance index given by the solution of the implicit Equation (3). We also use the same data as examples of one-period gambles to compare the AS performance index, given by the solution of the implicit Equation (2), with the AS performance index for multi-period gambles, given by the solution of the implicit Equation (3). Furthermore, we compare the two different AS performance indexes with the Sharpe ratio computed from data. To derive the AS performance index for multi-period gambles, we employ several parametric models under the maintained assumption that the underlying stochastic process of multi-period gambles follows a normal mixture process with time-varying volatility of GARCH families. We follow Alexander and Lazar (2009); Haas et al. (2004); and Xu and Wirjanto (2010) to assume that the return Xtof an asset is given by Xt=et(4) where etfollows a mixture of Knormal distributions with a time-varying volatility process et|It−1∼πkN(µk,σ2 k,t)(5) for t= 1, ··· , T and k= 1, ··· , K , where N(µk , σ2 k,t) denotes normal distribution with mean µk and variance σ2 k,t , It−1 is the information set up to time t− 1, 0 ≤πk≤ 1, and ∑K k=1πk= 1. We assume that the conditional variance of the k-th component follows three possible processes; (1) GARCH(1,1) process σ2 k,t=ωk+αke2 t−1+βkσ2 k,t−1(6) (2) Asymmetric GARCH(1,1) (AGARCH(1,1)) process σ2 k,t=ωk+αk(et−1−λk)2+βkσ2 k,t−1(7) (3) GJR(1,1) process (the model based on Glosten et al. (1993)) σ2 k,t=ωk+αke2 t−1+λkd− t−1e2 t−1+βkσ2 k,t−1(8) where d− t= 1 if et< 0 and 0 otherwise, component conditional variances depend on the previous innovation et−1as well as their own previous conditional variances. We make component conditional variances not dependent on the previous conditional variances of other components. Then, the conditional mean, variance, skewness, and kurtosis of Xt given the information set up to time t−1 are given respectively by µ= K ∑ k=1 πkµk σ2 t= K ∑ k=1 πk(σ2 k,t+µ2 k)−µ2 τt=1 σ3 t K ∑ k=1 πk(µk−µ)h3σ2 k,t+ (µk−µ)2i(9) κt=1 σ4 t K ∑ k=1 πkh3σ4 k,t+6(µk−µ)2σ2 k,t+ (µk−µ)4i. When Xt follows the above normal mixture process with time-varying volatility of GARCH families, the following equality holds for E[exp(−PAS(g)·gt)] in the implicit Equation (3) of the AS performance index: E[exp(−PAS(g)·gt)] = K ∑ k=1 πkexp(−µkPAS(g) + σ2 k,tPAS(g)2/2)(10) J. Risk Financial Manag. 2020,13, 288 6 of 18 since the moment-generating function (MGF) E[exp(sY)] of a random variable Yis given by exp(µs+σ2s2/2)(11) when Y follows normal distribution N(µ , σ2) . Notice E[exp(−PAS(g)·gt)] is, besides the minus sign, nothing but the MGF of gtas a function of PAS(g). When a multi-period gamble g= (g1 , g2 , ··· , gT) follows the normal mixture process with the above GARCH families, the AS performance index for the gamble g= (g1 , g2 , ··· , gT) is given by the unique solution PAS(g)to the implicit equation T ∑ t=1 ρt−1K ∑ k=1 πkexp(−µkPAS(g) + σ2 k,tPAS(g)2/2) = T ∑ t=1 ρt−1. (12) where ρ∈(0, 1)is a discount factor. In this study, we seek to obtain the AS performance index for multi-period gambles empirically. Providing a sufficient condition for existence of the AS performance index for multi-period gambles as in that for one-period gambles given by Aumann and Serrano (2008); Homm and Pigorsch (2012b); and Sculze (2014) is beyond the scope of this study. To obtain the AS performance index for multi-period gambles, we first estimate the parametric models for the underlying stochastic model of the normal mixture process with time-varying volatility of GARCH families. To estimate the parametric models, we use an empirical characteristic function (ECF) approach of Xu and Wirjanto (2010) which has several advantages as a method of estimating the parametric models: a closed-form objective distance function is available, the estimator has strong consistency and asymptotic normality, and the characteristic function is always uniformly bounded, unlike the likelihood function which is not always bounded over its parameter space (cf. Xu and Wirjanto 2010). We employ the continuous empirical characteristic function (CECF) approach by Xu and Wirjanto (2010) to estimate the AS performance index when the underlying stochastic process is given by the normal mixture process with time-varying volatility of GARCH families. The CF associated with Equations (4) and (5) is defined by Ct(r,θ) = E[eirXt] = K ∑ k=1 πkexp iµkr−1 2σ2 k,tr2(13) where i=√−1 and θdenotes the set of parameters in the model. The ECF of the above equation is given by Ct(r,Xt) = exp(irXt). (14) Then we consider the following distance measure given by Dt(θ;Xt) = Z|Ct(r,Xt)−Ct(r,θ)|2exp(−br2)dr. (15) We have the following result for the closed-form expression of the above distance function Dt(θ;Xt). J. Risk Financial Manag. 2020,13, 288 7 of 18 Proposition 3 (Proposition 1 of Xu and Wirjanto (2010)).If the return Xtis generated from Equations (4) and (5) and the distance measure under the CECF is given by Equation (15), then the closed-form-expression for the distance measure Dt(θ;Xt)is given by Dt(θ;Xt) = rπ b+ K ∑ k=1 π2 ksπ b+σ2 k,t −2 K ∑ k=1 πksπ 1 2σ2 k,t+bexp −(Xt−µk)2 4b+2σ2 k,t!  +2∑ k6=h πkπhsπ b+1 2(σ2 k,t+σ2 h,t) ×exp(−(µk−µh)2 4b+2(σ2 k,t+σ2 h,t)). (16) We remark the conditional variance σ2 k,t of the k-th component in the closed-form-expression given above can be any of the three possible processes of GARCH families given above. In other words, the closed-form-expression (16), originally for the standard GARCH models, continues to hold for other forms of GARCH families such as AGARCH and GJR models. We employ b= 1 when we implement estimation by minimizing the closed-form expression as in Xu and Wirjanto (2010). The CECF estimation of the model is to minimize D(θ) = ∑T t=1Dt(θ ; Xt) with respect to the set of unknown parameters in the model. The following result holds for the asymptotic normality result. Proposition 4. √T(ˆ θ−θ) =⇒N(0, Λ−1ΩΛ−1)(17) where ˆ θ denotes the estimator by the CECF approach, =⇒ denotes convergence in distribution, Λ=Eh∂2D(θ) ∂θ∂θ0i , and Ω=Eh∂D(θ) ∂θ ∂D(θ) ∂θ0i. See Heathcote (1977) for the proof of the above proposition. 4. The Empirical Estimation Results In this section, we estimate the AS performance index for multi-period gambles by the CECF approach described in the previous section when the underlying stochastic process is assumed to be the normal mixture process with time-varying volatility of GARCH families. We employ a selection of U.S. stock return data to estimate the AS performance index. We estimate the AS performance index for multi-period gambles as well as for one-period gambles and compare the two AS performance indexes to find out the consequences of the assumption for multi-period gambles. The AS performance index for one-period gambles is obtained as a parametric MLE assuming the underlying stochastic process follows the normal mixture process with time-invariant volatility, which is a special case of the normal mixture process with time-varying volatility of GARCH families, i.e., the assumption of the underlying stochastic process for the AS performance index for multi-period gambles. We use the two market indexes of the Dow Jones Industrial Average (DOW) and Nasdaq Composite Index (NASDAQ) and individual stocks of Johnson and Johnson (JNJ), Amazon, and Microsoft as a selection of U.S. stocks. We employ daily return data of these stocks from January 2, 2008, till April 28, 2017. We show the Sharpe ratio for these data in addition to the AS performance index for multi-period gambles as well as for one-period gambles. The Sharpe ratio is computed directly from data. The risk-free rate is obtained from the Treasury bill rate data, downloaded from Ken French’s homepage. We first provide summary statistics for the stock return data in Table 1. Summary statistics are mean, standard deviation (s.d.), skewness, and kurtosis. Mean ranges from 0.027 in DOW to 0.129 in J. Risk Financial Manag. 2020,13, 288 8 of 18 Amazon. Standard deviation ranges from 1.054 in Johnson & Johnson to 2.485 in Amazon. Therefore, Amazon is a stock with the highest mean and the highest risk. All the stocks are positively skewed with the exception of NASDAQ. All the stocks have heavy tails compared to the normal distribution. Table 1. Summary Statistics of the Stock Returns. Table 1presents summary statistics of the stock return data we study in this paper, i.e., mean, standard deviation (s.d.), skewness, and kurtosis. Name Mean s.d. Skewness Kurtosis DOW 0.027 1.221 0.157 13.836 NASDAQ 0.045 1.407 −0.071 10.451 JNJ 0.044 1.054 0.690 16.582 Amazon 0.129 2.485 0.977 14.435 Microsoft 0.044 1.793 0.467 13.784 JNJ stands for Johnson & Johnson. We follow Engle and Ng (1993) to test serial correlation in levels and squares of stock return data and to examine if the value of et−1 influences current volatility. Table 2provides results of Ljung-Box statisics 2 of serial correlation of the twelfth-order for levels and squares of stock return data as well as results of the sign bias test statistic, negative size bias test statistic, positive size bias test statistic, and joint test statistic described in Engle and Ng (1993). The sign bias test statistic, negative size bias test statistic, and positive sign bias test statistic given in Engle and Ng (1993) are respectively the t-test statistic of the explanatory variable d− t−1 , d− t−1et−1 , and d+ t−1et−1 in the regression equation of the normalized residual as the dependent variable where d− t−1= 1 if et−1< 0 and 0 otherwise, d+ t−1=d− t−1+ 1, and the normalized residual is the residual divided by the conditional standard deviation estimate. The joint test statistic is the Lagrange multiplier (LM) test statistic for adding the three variables of the sign bias, negative size bias, and positive sign bias in the regression equation of the normalized residual as the dependent variable. Table 2shows all the stocks are serially uncorrelated in levels but serially highly correlated in squares of stock returns, which are conformable with the stylized facts of financial data. With respect to the effects of the value of et−1 on current volatility, the negative size bias test is always highly significant, which is the same as in Engle and Ng (1993) for the Japanese stock index data in their study. The sign bias test statistic is positive and highly significant for the two indexes but insignificant for two individual stocks of Amazon and Microsoft. Later, we will see the best model in Amazon and Microsoft are GARCH(1,1) three components models, the most complicated models, where the effect of the negative sign of et−1 being not in simple forms may be due to this insignificant result. The positive size bias test is not significant for the two indexes, which is the same as in Engle and Ng (1993) for the Japanese stock index in their study. However, it is positive and significant for the three individual stocks. Therefore, the negative size bias test is significant while the positive size bias test is not significant in the two indexes of the DOW and NASDAQ, indicating the effect of et−1 on current volatility is asymmetric in the DOW and NASDAQ. On the other hand, the negative size bias test and the positive size bias test are both significant in Johnson & Johnson, Amazon, and Microsoft, indicating the effect of et−1 on current volatility is symmetric in the three individual stocks. The joint test is highly significant for all the stocks. 2 The Ljung-Box statistics we use here are modified ones by Diebold (1988) who corrected the original Ljung-Box statistic which is known to reject the null hypothesis too often. J. Risk Financial Manag. 2020,13, 288 15 of 18 mean and highest risk, which causes it to take negative values often, which makes its AS performance score low. On the other hand, stable stocks such as Johnson & Johnson perform fairly well, which are rated highly by the AS performance index (cf. Hodoshima 2019). Table 10. The Sharpe Ratio in Ascending Order for the Stock Returns. Name Sharpe Ratio DOW 0.021 Microsoft 0.024 NASDAQ 0.031 JNJ 0.041 Amazon 0.051 JNJ stands for Johnson & Johnson. Table 11. The AS Performance Index for One-period Gambles in Ascending Order for the Stock Returns. Table 11 presents the AS performance index for one-period gambles assuming the underlying stochastic process to be a normal mixture process with time-invariant volatility. Name AS Index Microsoft 0.027 DOW 0.037 Amazon 0.042 NASDAQ 0.046 JNJ 0.079 JNJ stands for Johnson & Johnson. To compare the fitness of the best model in the models for one-period gambles and multi-period gambles, we provide scores of the BIC in the best model of the normal mixture process with time-invariant volatility in Table 12. When we compare scores of the best BIC in the models for one-period gambles with those for multi-period gambles, given in Tables 3–7, we can see the best models for multi-period gambles dominate those for one-period gambles. Hence, the model selection criterion of the BIC uniformly chooses the model for multi-period gambles as compared to the model for one-period gambles in all the stocks we consider. Table 12. The BIC of the Best Model of One-period Gambles for the Stock Returns. Name BIC DOW 6822.029 NASDAQ 7667.211 JNJ 6379.585 Amazon 10308.100 Microsoft 8805.842 JNJ stands for Johnson & Johnson. We then show the AS performance index for multi-period gambles in Table 13. We set the discount factor ρ to be 0.01, 0.05, 0.10 in annual rate. We show the AS performance index for the best model by the BIC as well as the second best model by the BIC to see how the result changes as the model changes in all the stocks we consider. The results show the AS performance index for multi-period gambles tends to be larger than that for one-period gambles when the AS performance index for one-period gambles is larger than the Sharpe ratio. The AS performance index for one-period gambles is larger than the Sharpe ratio in four stocks out of the five stocks we examine in this paper, i.e., DOW, NASDAQ, Johnson & Johnson, and Microsoft. One exception of the above property, i.e., the AS performance index for multi-period gambles tends to be larger than that for one-period gambles when the AS performance index for one-period gambles is larger than the Sharpe ratio, is Johnson & Johnson J. Risk Financial Manag. 2020,13, 288 16 of 18 where the AS performance index for multi-period gambles is similar to that for one-period gambles in the best model of the GARCH two components model. However, the AS performance index for multi-period gambles is distinctively larger than that for one-period gambles in the second-best model of the GJR one component model in Johnson & Johnson. We remark the same is true in the third-best model of the AGARCH one component model in Johnson & Johnson where the AS performance index for multi-period gambles is 0.101, 0.098, and 0.093 respectively when ρ= 0.01, 0.05, and 0.10. The AS performance index for one-period gambles is smaller than the Shape ratio only in Amazon among the five stocks we examine in this paper. The AS performance index for multi-period gambles is even smaller than that for one-period gambles when the AS performance index for one-period gambles is smaller than the Sharpe ratio. This occurs in the case of Amazon. When we compare the AS performance index with the Sharpe ratio, a favorable evaluation score tends to become even better in multi-period gambles than in one-period gambles while an unfavorable evaluation score tends to become even worse in multi-period gambles than in one-period gambles. This property is new, although our results are limited since we only examine a handful of stocks. Whether this property is only empirical or an intrinsic property of the relationship between the AS performance index and Sharpe ratio is beyond the scope of the present study and is left as a future research topic to be studied. Table 13. The AS Performance Index for Multi-Period Gambles for the Stock Returns. Name ρ= 0.01 ρ= 0.05 ρ= 0.1 DOW MinBIC AGARCH-2 0.051 0.048 0.044 2ndMinBIC GJR-1 0.068 0.063 0.058 NASDAQ MinBIC AGARCH-2 0.070 0.066 0.062 2ndMinBIC GJR-1 0.089 0.084 0.078 JNJ MinBIC GARCH-2 0.080 0.078 0.074 2ndMinBIC GJR-1 0.100 0.096 0.092 Amazon MinBIC GARCH-3 0.025 0.025 0.024 2ndMinBIC AGARCH-3 0.036 0.034 0.032 Microsof MinBIC GARCH-3 0.047 0.046 0.044 2ndMinBIC AGARCH-3 0.031 0.029 0.028 JNJ stands for Johnson & Johnson. Table 13 presents the AS performance index for multi-period gambles in the best model and second best model with respect to the BIC of the three classes of GARCH(1,1) families. MinBIC and 2ndMinBIC stand for respectively the best model and second best model with respect to the BIC. In the table, AGARCH-2 and GJR-1 in the DOW denote respectively an AGARCH(1,1) two components model and a GJR(1,1) one component model. Other names of these models in the table are similarly defined. 5. Concluding Comments In this paper, we presented an empirical study of the AS performance index for multi-period gambles under the setup given by Kadan and Liu (2014) where the utility function of multi-period gambles is time-separable with a discount factor. We obtained the parametric estimates of the AS performance index for multi-period gambles assuming the underlying stochastic process of returns to follow the normal mixture process with time-varying volatility of GARCH families. Our empirical study of the AS performance index is the first empirical study of the AS performance index for multi-period gambles. We compared the AS performance index for multi-period gambles with that for one-period gambles as well as the Sharpe ratio obtained from data. Estimates of the AS performance index for one-period gambles are obtained assuming the underlying stochastic process of returns to follow the normal mixture process with time-invariant volatility. J. Risk Financial Manag. 2020,13, 288 17 of 18 Our results show the following property of the estimates of the AS performance index for multi-period gambles. Stocks with higher AS performance index scores for one-period gambles than the Sharpe ratio are evaluated even better in the AS performance index for multi-period gambles. On the other hand, stocks with lower AS performance index scores for one-period gambles than the Sharpe ratio are evaluated even worse in the AS performance index for multi-period gambles. Our results are obtained using a selection of U.S. stocks and show obtaining the AS performance index for multi-period gambles is not difficult under a reasonable and tractable assumption of the normal mixture process with time-varying volatility of GARCH families. Therefore, the AS performance index for multi-period gambles is a practical tool for evaluation of multi-period gambles, which opens up a new way of evaluating various dynamic gambles. Obtaining appropriate performance measures is essential to investors, financial managers, and regulators, we believe our approach of the AS performance index for multi-period gambles is widely useful to these audiences. Exploring the AS performance index for multi-period gambles with the different class of assets, projects, cash flows, etc. is important and interesting and left as a future research topic. Author Contributions: Conceptualization, J.H. and T.Y.; Methodology, J.H. and T.Y.; Software, J.H. and T.Y.; Validation, J.H. and T.Y.; Formal Analysis, J.H.; Investigation, J.H. and T.Y.; Resources, J.H. and T.Y.; Data Curation, T.Y.; Writing—Original Draft Preparation, J.H.; Writing—Review & Editing, J.H. and T.Y.; Visualization, J.H. and T.Y.; Supervision, J.H.; Project Administration, J.H.; Funding Acquisition, J.H. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by JSPS KAKENHI Grant Number JP17K03667. 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